Holographic eﬀective ﬁeld theories: a case study

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Universit`

a degli Studi di Padova

Dipartimento di Fisica e Astronomia

Corso di Laurea Magistrale in Fisica

Holographic effective field

theories: a case study

Laureando:

Riccardo Antonelli

Relatore: Luca Martucci

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Abstract

The identification of the low-energy effective field theory associated with a given mi-croscopic strongly interacting mimi-croscopic theory constitutes a fundamental problem in theoretical physics, particularly challenging when the theory is not sufficiently constrained by symmetries. Recently, a new approach has been proposed, which addresses this problem for a large class of four-dimensional minimally supersym-metric strongly coupled field theories, admitting a dual weakly coupled holographic description in string theory. This approach includes a precise prescription for the derivation of the associated effective theories through holography. The aim of this thesis is to explore these techniques by specializing them to a specific model whose effective theory has not been investigated before.

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Introduction

Strongly-coupled quantum field theories represent canonical examples of physical systems whose study is extremely challenging. Even the question of the mere existence of any interacting QFT in four dimensions from a formal standpoint has not been settled. In addition to this, strong couplings are not amenable to the tools of perturbation theory. The interest in this class of theories stems actually from practical considerations - many of them represent realistic models for physical phenomena, e.g. the theory of strong interaction.

A subset of questions concerns whether a given strongly-interacting theory is des-cribed at low energy by an effective local field theory, and if so, what are its degrees of freedom and their precise dynamics. Often, part of the structure of the effective theory is constrained by symmetries, but no general method exists to fix it comp-letely. Recently[24], a novel approach for determining the effective Lagrangian was introduced that makes use of tools from an apparently unrelated area of physics: string theory.

It is remarkable that string theory was originally conceived as a description of hadronic physics, so a low-energy effective theory for what ultimately turned out to be a gauge theory, QCD. When string theory was found to have unsuitable quali-ties for this application, it was replaced by the theory of quantum chromodynamics - however it also proved to be effective for solving a seemingly unrelated problem of fundamental physics: quantizing gravity. Since then, string theory blossomed into a vast and rich field reaching into numerous areas of mathematics and physics, and of course a candidate for a “Theory of Everything” describing the entirety of fundamental physics.

Among the most unexpected discoveries in strings, made decades after their concep-tion, is a series of unusual exact equivalences between string theories set in parti-cular ten-dimensional backgrounds and four-dimensional gauge QFTs. More gen-erally, one finds families of exact equivalences between local quantum field theories and higher-dimensional theories containing gravity, which are termed “holographic”.

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2 CHAPTER 1. INTRODUCTION This explains the original partial success of strings in modeling strong interactions, assuming that some or perhaps most gauge theories have or can be approximated as having a holographic description as a ten dimensional theory involving strings. This roundtrip has therefore brought strings back to strongly-coupled gauge theo-ries. Various aspects, qualitative and most importantly quantitative, of QFTs can be studied directly by means of their holographic string dual, a gravitational theory, if it exists.

The first and most important case of such a duality[19] equates IIB superstring theory set on the background

AdS5×S5 (1.1)

(the “bulk”) with maximally supersymmetric Yang-Mills theory on R1,3 (the “boun-dary”), which is a conformal field theory. The denomination of “AdS/CFT” (anti-de Sitter / conformal field theory) correspon(anti-dence for holographic dualities stems from this (even though cases either with no AdS geometry or not conformal are known). One direction in which to generalize this construction is to replace S5 _with

other compact 5-manifolds Y5. This yields dualities involving more complex and

interesting field theories, less constrained by symmetries; therefore this thesis will be focused on this class of correspondences.

Actually, since string theory in general is very challenging to study, AdS/CFT only becomes truly useful in terms of describing the dynamics of the CFT if string theory can be approximated by a weakly coupled effective field theory of its own, supergra-vity. This limit corresponds to the CFT being strongly-coupled and having a large number of colours. Therefore the regime accessible through holography is precisely the strongly-coupled region where the gauge theory would be normally impossible to investigate.

Recently, a novel approach was introduced[24] for determining the effective theory to such duals of AdS5×Y5. A procedure for identifying the degrees of freedom

and the Lagrangian for the effective low-energy theory for a class of gauge theories with AdS5×Y5 holographic duals is provided, by expanding the supergravity action

on the dual bulk geometry. Interestingly, these are somewhat special in that they include models of minimal supersymmetry (N = 1), which makes for more realistic but by converse less constrained theories than typical holographic field theories, with higher supersymmetry. The ability to pinpoint the exact effective Lagrangian is then particularly noteworthy.

The original contribution in this work is the specialization of this construction to a specific field theory, the Y2,0 _{theory, a strongly-coupled superconformal quiver}

theory for which we will therefore fix the exact effective Lagrangian, entirely through the geometry of the relative string background. This will require necessarily the determination of the general Calabi-Yau deformation of the background in complex

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3 coordinates, which was not known previously.

This thesis will be structured as follows. We will first provide a general introduction to IIB superstring theory, D-brane stacks on cones and the resulting gauge field the-ories, and holography. Then, we will summarize the relevant results and techniques from [24]. Finally, we will present a complete parametrization of the geometry of the Y2,0 theory and will apply those results and techniques to identify the exact effective Lagrangian of the field theory.

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Chapter 2

IIB superstrings and branes

String theories are quantum theories involving 1-dimensional dynamical objects, the fundamental strings. Therefore, they can be regarded as a generalization of systems of 0-dimensional quantum particles, such as quantum field theories. This apparently innocuous modification results in amazing depth and complexity of the structure of the resulting theories. The celebrated inclusion of Einstein gravity is only a small part of the large diversity of phenomena.

The downside is a significant difficulty in probing the general structure of these theories. Only the behaviour in particular regimes is known, while the overall theory interpolating between these limiting cases is mostly unknown, or at the very least impossible to formulate. Consequently, we will only attempt an introduction to the aspects of string theory relevant to our purposes, namely the conditions under which strings can be approximated by an effective field theory (and which one), and stable non-perturbative states known as D-branes.

Our interest will be directed towards string theories involving supersymmetry, and in particular a specific variety known as type IIB string theory, set in ten dimensions, which will always be involved in the holographic dualities we will study. In fact, IIB will be both the background theory employed to construct the holographic duality, and one side of the duality itself. Thus, we now provide a summary starting from perturbative superstring theory to present some general aspects of type-IIB strings, including the corresponding effective field theory and D-branes.

2.1 Superstring theory

String theory either does not admit a nonperturbative Lagrangian formulation, or this formulation is unknown. An action functional can only be written upon choosing a perturbative vacuum; since we anticipate a string theory must include gravity, a choice of vacuum will also require a choice of background metric - in the simplest case Minkowski spacetime. The configuration of a string moving in this spacetime

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6 CHAPTER 2. IIB SUPERSTRINGS AND BRANES (the target M ) is then given by specifying the two-dimensional submanifold (the worldsheet W1) it traces in it1, the worldsheet. In essence, this coincides with

providing an embedding of the worldsheet

Xµ(τ, σ) : W1→ M (2.1)

then quotiented under diffeomorphisms of the coordinates τ, σ on W1.

With a given choice of background metric the most natural action for a string is the Nambu-Goto action, the worldsheet area:

SN G[X] = −T Z W1 d volh = −T Z W1 d2σ√−h (2.2) where hab = ∂x µ ∂σa∂x ν

∂σbGµν is the induced metric on the worldsheet from the target

space metric Gµν under the embedding given by Xµ. T instead is a dimensionful

constant called the string tension; in fact it is the only free parameter in string theory. We will also often refer to the entirely equivalent quantity α0, the Regge slope.

T = 1

2πα0 (2.3)

The Nambu-Goto action is very difficult (if not impossible) to quantize. It proves much easier to switch to the classically equivalent Polyakov action:

SB[X, g] = − T 2 Z W1 d2σ√−ggab∂aXµ∂bXνGµν (2.4)

where now gab is an independent auxiliary field, not the induced metric from the

Xµ. The equivalence is readily shown by computing the classical equation of motion for gab and substituting back into SB to recover SN G.

There are essentially two2 _{different sensible choices for the topology of W}

1: either a

cylinder, with σ being the periodic variable running around, or a strip, so that σ is limited to an interval [0, σ1]. These are respectively the closed and open string. The

former is always a closed loop at any given instant in time. The open string instead has two endpoints for which we have to fix boundary conditions. One could choose between either Neumann boundary conditions, meaning

∂Xµ ∂σ σ=0,σ1 = 0 (2.5)

1_{We note the worldsheet is just the obvious generalization of the concept of worldline of a particle}

to the case of 1-dimensional strings.

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2.1. SUPERSTRING THEORY 7 which is just the constraint that no momentum flows out of the string endpoints, or Dirichlet boundary conditions, which fix

Xµ|_σ=0= X₀µ, Xµ|_σ=σ

1 = X

µ

σ1 (2.6)

where X₀µ, Xσµ1 are constants, essentially forcing the string endpoints to a specific

spacetime point. In general one could mix p + 1 Neumann conditions and D − p − 1 Dirichlet conditions for different values of µ, so that the endpoints are constrained to a p-dimensional submanifold in space, and can move freely within it. Dirichlet conditions evidently break the symmetries of the target spacetime (Poincar´e if we choose a Minkowski background) as they specify a preferential frame and submani-fold; this symmetry will be recovered when it is recognized that the p-dimensional submanifold to which open strings attach is actually a dynamical object, a Dp-brane (D alluding to Dirichlet). We will return to D-branes after studying the string spec-trum.

The Polyakov action displays invariance under worldsheet diffemorphisms

σa→ σ0a(σa) (2.7)

and Weyl transformations:

gab → eφ(σ)gab (2.8)

and thus perturbative string theory is naturally a two-dimensional conformal field theory. These symmetries must be quotiented out someway on quantization. The most straightforward way is to eliminate them by fixing a particular gauge and then quantizing (canonical quantization). The three symmetry generators can kill the three degrees of freedom in the metric to fix it to the 2D Minkowski: gab= ηab. We

get SB = − T 2 Z d2σ∂aXµ∂aXµ (2.9)

where indices are raised with ηab.

The theory described so far would be what is known as bosonic string theory. There are two issues with bosonic strings: the first is the presence of tachyons in both the open and closed string spectrum, i.e. some string modes will have a negative mass squared, signaling an instability of our choice of perturbative vacuum. The second is that, as the name suggests, there are exclusively bosons in the spectrum, which makes it unsuitable at the very least for phenomenological application. A modification to include supersymmetry can be performed to solve both of these

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8 CHAPTER 2. IIB SUPERSTRINGS AND BRANES issues and produces a set of string theories with fermions and without tachyons, the superstrings. We will in particular sketch the path from bosonic string theory to the so-called type II superstrings, two slightly different theories IIA and IIB.

There are at least two different approaches to introducing supersymmetry into a string theory. The path followed by the RNS (Ramond-Neveu-Schwarz) formalism is to impose SUSY at the worldsheet level; explicitly, adding fermions ψµ to act as superpartners to the bosons Xµ. We follow the derivation in [4]. The action is extended to S = SB+ SF = − T 2 Z d2σ ∂aXµ∂aXµ+ ¯ψµρa∂aψµ (2.10)

where the ρ1,2 are two-dimensional gamma matrices satisfying the Clifford algebra n

ρa, ρb o

= 2ηab (2.11)

The spinors’ equation of motion, the Dirac equation, is actually the Weyl condition in two dimension. This brings the real degrees of freedom in the spinor for each µ from 4 to 2. Recalling that in (2 mod 8) dimensions there exist Weyl-Majorana spinors satisfying both the Weyl and Majorana conditions, imposing the latter on ψ halves again the on-shell polarizations to 1. Thus we have a match between bosonic and fermionic degrees of freedom. It can be proven the theory above is indeed worldsheet supersymmetric.

To quantize canonically, we introduce canonical commutation/anticommutation re-lations:

[Xµ(σ), Xν(σ0)] = ηµνδ2(σ − σ0) {ψµ(σ), ψν(σ0)} = ηµνδ2(σ − σ0) (2.12)

Note the X0 and ψ0 would create negative norm states, but these modes are elimi-nated by resorting to superconformal invariance. Classically this symmetry imposes the stress-energy tensor Tµν and the supercurrent J_αa vanish; imposing that in the quantum theory they annihilate physical states yields the restriction that removes the longitudinal ghosts from the spectrum. These take the name of super-Virasoro constrain.

Then the procedure for building the string spectrum is to expand the classical solu-tions in terms of Fourier modes, identify creators and destructors, and then select the states of the Fock basis that satisfy the super-Virasoro constraints. There are various ways to proceed at this point; the simplest and perhaps the most inelegant is light-cone quantization, which we will refer to. Essentially, a subset of D − 2 transverse directions i = 2, . . . , D − 1 are selected and only Xi and ψi are made to correspond to operators, and the remaining longitudinal fields are to be determined from the

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2.1. SUPERSTRING THEORY 9 former by the classical constraints. This procedure renders Lorentz-invariance non-manifest - we will verify indeed it is actually recovered a posteriori. A different common approach, more rigorous, takes the form of a BRST quantization, akin to that employed in Yang-Mills theories, by exchanging superconformal gauge inva-riance with the introduction of a series of Faddev-Popov ghosts. The final quantum theories are identical and the choice of quantization is conventional.

Boundary conditions for ψµ for an open string can actually be satisfied in two different by imposing periodicity or antiperiodicity, giving rise to the NS (Neveu-Schwarz) and R (Ramond) sectors, built over two grounds |0i_{N S} and |0i_R. Closed strings have four: |0i_{N S−N S}, |0i_R−R, |0i_{R−N S}, |0i_{N S−R}corresponding with different choice periodicity conditions for left and right-movers.

2.1.1 Open strings

It can be shown that while the NS ground |0i_{N S} is unique, and thus a spacetime scalar, |0i_R is eight-fold degenerate and this 8-plet transforms under the spinor representation of transverse SO(8) - in other words, it is a spacetime spinor. In particular, it is a chiral Weyl-Majorana spinor, so it can be taken to be either of positive or negative chirality, choices we will denote as |+ia_R, |−i_R˙a, with a, ˙a = 1, . . . , 8 the spinor index.

The spectrum is built by acting on one of the grounds with bosonic and fermionic creators, to obtain states of higher and higher mass. For the NS sector, there are bosonic creators ai†n (n ≥ 1) and fermionic bi†r (r positive half-integer), and the mass

of the excited string is given by:

α0M2 = ∞ X n=1 n ai†_nai_n+ ∞ X r=1/2 rbi†_rbi_r−1 2 (2.13)

while for the R sector the fermionic creators are replaced by the integer-indexed d†in:

α0M2= ∞ X n=1 n ai†_nai_n+ ndi†_ndi_n (2.14)

The i indices here are target spacetime transverse indices, i = 1, · · · , 8. Therefore each creator increases the spacetime spin of the string by one unit. We conclude the NS sector contains only spacetime bosons, and the R sector only spacetime fermions. The constant shift of −1/2 in (2.13) (and of 0 in (2.14)) actually results from an ordering ambiguity of the creators and destructors a†n, an (omitting spacetime

in-dices for now) which introduces an arbitrary constant shift in the Hamiltonian upon quantization. If we want a consistent quantum theory with Lorentz invariance, this shift in the NS sector is fixed to −1/2; this is because for example the state

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10 CHAPTER 2. IIB SUPERSTRINGS AND BRANES

bi†_1/2|0i_{N S} (2.15) is a vector with (D −2) polarizations, and thus must be massless. However, it can be shown that this shift is also related to the number of spacetime dimensions D, so that a condition restricting D to a particular value can be found. An intuitive and perhaps heuristic explanation of this fact is as follows: the energy shift should be equal to the sum of the zero-point energies (ZPE) of the infinite harmonic oscillators, bosonic for n = 1, 2, . . . and fermionic for r = 1/2, 3/2, . . . We recall bosonic / fermionic QHOs have Hamiltonians H = ω a†a + 1 2 H = ω b†b − 1 2 (2.16)

Then for a given value of i the sum of all the ZPEs of the oscillators in the NS sector would be E0= X n∈N n 2 − X r∈N+1₂ r 2 = 1 4 X m∈N (−1)mm (2.17)

The sum −1 + 2 − 3 + . . . is evidently divergent; we assume it is admissible to replace it with its ζ-regularized value3 of −1/4. Therefore the ZPE per transverse direction is E0 = −1/16; insisting the total ZPE is equal to −1/2 results in

−D − 2 16 = −

1

2 ⇒ D = 10 (2.18) While this argument is not completely rigorous, it correctly identifies the existence and value of the so called critical dimension D = 10 for superstring theories. In a more formal setting (e.g., in BRST quantization) it can be shown that the classical Weyl symmetry of the action is spoiled by quantization and a conformal anomaly arises; this anomaly can be proven to cancel4 only for D = 10.

It may worry that the mass-shell formula above assigns a negative mass-squared to the NS ground, which is therefore a tachyon. In addition, it is the only tachyon, meaning this theory is not spacetime supersymmetric. We will see in the next section how this state is actually removed and target supersymmetry recovered. For now, we note the only massless states are

3_{The evaluation by analytic continuation of the divergent series is easily computed by considering}

1 − 2x + 3x2− . . . = (1 + x)−2

, converging for |x| < 1. Setting x = 1, right on the edge of the disk of convergence, yields the desired result.

4

it is possible to have the conformal anomaly cancelled by the introduction of additional fields if D < 10, resulting in non-critical string theories. These have properties that make them unsuitable for our applications, however, and we will ignore them.

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2.1. SUPERSTRING THEORY 11

bi†_1/2|0i_{N S} |+i_{N S} (2.19) while the rest of the tower of states have string scale-large ∼ (α0)−1/2 masses. Our interest in massless modes stems from the fact that in a low-energy (α0 → 0) the strings can be approximated as pointlike particles (as their typical size ls ∼

√ α0₎

and their quantum theory as the corresponding field theory, with a field for each massless string mode, as the massive modes have decoupled. Such an effective low-energy theory will be described in section 2.3.

The first of the two states in (2.19) is a massless spin-1 boson, so it must be a photon associated with a U (1) gauge theory. The latter is its spin-1/2 superpartner, a photino. It must be noted that what was described up to now holds for the directions in which the string endpoints are free to move, hence those for which Neumann conditions are imposed. As we have anticipated open strings in general end on Dp-branes and D−p−1 directions are actually constrained by Dirichlet conditions so as to keep the endpoints on the brane; the conclusion is the gauge interaction this massless string mode mediates is actually confined to the p-dimensional volume of the brane.

2.1.2 GSO projection

The construction above does not define a consistent theory. This is in part because it is not spacetime supersymmetric, an essential requirement considering that, as will be seen shortly, the closed string spectrum includes a gravitino (a massless spin-3/2 state) which must be associated with local supersymmetry. A procedure known as the Gliozzi, Scherk, Olive (GSO) projection solves this issue and in addition also eliminates the tachyonic state |0i_{N S}, to end up with a consistent quantum theory. The following operator is introduced, acting on the NS sector as

G = (−1)1+Prb i†

rbir _{= (−1)}F +1ˆ _(2.20)

and on the R sector as

G = Γ11(−1) P rd i† rdir _{= Γ} 11(−1) ˆ F _(2.21) ˆ

F is the worldsheet fermion number, and Γ11 = Γ0· · · Γ9 gives the chirality of the

state.

Then the spectrum is projected into the G = 1 subspace for the NS sector, and into G = ±1 (either choice works) for the R sector. These two choices correspond to keeping either |+ia_R or |−i_R˙a respectively and discarding the other.

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space-12 CHAPTER 2. IIB SUPERSTRINGS AND BRANES time supersymmetric. The scalar tachyon |0i_{N S} in particular is eliminated, being G-odd.

2.1.3 Closed strings

The closed string spectrum, in somewhat poetic language, is the “square” of the open string spectrum. On closed strings, excitations are allowed to move in either clockwise or counterclockwise direction along the string, forming a left-moving and a right-moving spectrum5. As seen before the choice can be made for either NS or R boundary conditions, and this can be performed separately for left-movers and right-movers, giving four sectors. The GSO projection is performed separately on left and right movers, so that one is presented with the choice of the relative chirality of the two projections and so of the R grounds. These two possibilities will actually result in two different string theories. Choosing opposite chiralities gives type IIA strings, whose massless spectrum is given by

˜

bi†_1/2|0i_{N S}⊗ bj†_1/2|0i_{N S} (2.22) ˜

bi†_1/2|0i_{N S}⊗ |+ib_R (2.23) |−i_R˙a ⊗ bj†_1/2|0i_{N S} (2.24) |−i_R˙a ⊗ |+ib_R (2.25)

(the ∼ distinguishes creators/destructor for left movers from right movers). And IIB strings arise from equal chiralities:

˜

bi†_1/2|0i_{N S}⊗ bj†_1/2|0i_{N S} (2.26) ˜

bi†_1/2|0i_{N S}⊗ |+ib_R (2.27) |+ia_R⊗ bj†_1/2|0i_{N S} (2.28) |+ia_R⊗ |+ib_R (2.29)

So the massless spectrum is composed of 4 sectors of 64 physical states, two of them bosonic (NSNS, and RR) and the other fermionic (RNS and NSR). Massless states will correspond to fields in the supergravity approximation, in which the massive modes of the string decouple and the string theory is well described by the corresponding variety of 10D supergravity.

To present these states in a less opaque manner, consider for example the NSNS

5_{In the open string, the boundary conditions fixed the right-movers in terms of the left-movers}

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2.1. SUPERSTRING THEORY 13 mode (2.22), (2.26), equal in both theories. This is a tensor Hij; we can switch the transverse indices i, j to Lorentz indices µν provided we keep memory of the transversality constraint. Such a general tensor can be decomposed into Lorentz irreps as a symmetric traceless, antisymmetric and trace parts, as

Hµν = G(µν)+ B[µν]+ ηµνφ (2.30) The first of these is a transverse, massless, symmetric traceless rank-2 tensor, which means it must be a graviton. The presence of a field describing variations in the target spacetime metric from the chosen background metric signals that in general string theory will contain gravity. So in general string theory is a consistent theory of quantum gravity. We will describe the rest of the massless states better in the next sections.

The type II theories defined above are not the only consistent superstring theories. Three additional target-supersymmetric theories can be defined: type I, heterotic SO(32), and heterotic E8×E8, all set in 10 dimensions, for a total of five superstring

theories. Our interest will be mainly focused on the type II theories, however. A relevant point is that type II strings do not actually feature stable “free” open strings with all Neumann boundary conditions. Open strings can only appear at-tached to D-branes. (The discussion of section 2.1.1 was therefore mostly intended as preparatory to the introduction of the closed string spectrum). This means for example the closed-string spectrum above is sufficient to encompass the dynamics in the absence of D-branes, and that type II supergravity will result only from the massless closed string modes above.

2.1.4 Background fields, string coupling and loop expansion

It was already hinted that the massless string modes we found should give rise to fields in some limit. In particular the NS-NS closed string ground (2.26) is a spacetime rank-2 tensor, which can be split into symmetric, antisymmetric and trace parts.

Non-zero values of these fields could be incorporated back into the background the perturbative string is based on. In fact, since the symmetric tensor field above is actually the graviton, we already have: the Polyakov action (2.4) already includes a coupling of the string to the target background metric Gµν. This is just the

background value of the graviton.

The antisymmetric NS-NS Bµν field (equivalently, a 2-form B2 = 1₂Bµνdxµ∧ dxν),

called the Kalb-Ramond potential, is instead coupled to the fundamental string in a way that resembles the generalization of the coupling of a particle to the EM potential; a background value of Bµν would result in the addition to the action of a

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14 CHAPTER 2. IIB SUPERSTRINGS AND BRANES SB= T 2 Z W1 d2σεαβBµν(X)∂αXµ∂βXν ∝ Z W1 B2 (2.31)

this will result ultimately in a stringy generalization of electrodynamics with strings coupling to a field strength 3-form H3 = dB2.

Finally, strings will also couple to the last NS-NS closed string field, the trace scalar φ called the dilaton. The coupling to a background dilaton is of the form

Sφ= 1 4π Z W1 d2σ√g R(2)[g] φ(X) (2.32)

where R(2)[g] is the 2D Ricci scalar associated to g. Note that in the presence of a constant background dilaton φ(X) = φ, the integral above is simply the Euler characteristic χ(W1), an integer topological invariant, by the Gauss-Bonnet theorem.

χ has a simple expression in terms of the number of handles (the genus h), the number of boundaries nb and the number of cross-caps ncof the surface:

χ = 2 − 2h − nb− nc (2.33)

To understand the physical content of this contribution, we consider the simple case of orientable closed strings, and we imagine computing the amplitude of a process involving n external string states. Since only closed strings are involved, the only boundaries are the nb = n boundaries at infinity of the asymptotic states, and thus

nb is constant. Therefore the action from a constant dilaton background reduces to

Sφ= φχ = φ(2 − n − 2h) = const − 2hφ (2.34)

and the Euclidean path integral for the amplitude would take the form

A = Z

DXDψDg e−SP_e−2φ h[g] _(2.35)

apart from normalization, and the path-integral is over worldsheets that match the external states. The integral over metric structures R Dg splits into disconnected components indexed by the genus, so that

A = ∞ X h=0 Ah = ∞ X h=0 (eφ)−2h Z h DXDψDg e−SP _(2.36)

This is a loop expansion, since the genus h counts the number of virtual string loops; however it is also a perturbative expansion in the string coupling gs:= eφ, giving the

strength of a cubic string interaction “vertex”. Therefore, we come to understand that the applicability of the perturbative string theory, including what has been

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2.2. TYPE II SUPERGRAVITY AND D-BRANE CONTENT 15 introduced in this chapter so far, rests on the smallness of this string coupling gs.

It is worth of notice that this coupling is not an external dimensionless parameter of the theory (since string theory has none) but rather is related to the expectation value of a scalar field.

+ g_s2 + . . .

Figure 2.1: First terms in the loop expansion of a closed string four-point function. The worldsheets have been cut into “pair of pants” surfaces to count string interactions.

The perturbation series (2.36) is the stringy analogue of the QFT sum over Feynman diagrams. Its power comes from the fact that the genus-h contribution involves the calculation of a single diagram - the multiplicity of inequivalent Feynman graphs of a field theory (growing as ∼ eh) is then interpreted as the various inequivalent ways in which the unique worldsheet topology of genus h can degenerate to a diagram with pointlike particles and interactions as the string length is sent to zero.

Figure 2.2: The possible Feynman graphs from degeneration of the genus-0 worldsheet from figure 2.1

Open strings interactions are instead controlled by a different coupling go. Consider

the addition of an open string loop. This introduces two open string vertices and thus should result in a g2_osuppression. However, this operation results in the addition of a boundary and no change in genus, so ∆χ = −1. Therefore the suppression is also (eφ)−∆χ = gs and we find that

g_o2 ∼ gs (2.37)

2.2 Type II supergravity and D-brane content

At energy scales much lower than the string scale (α0)−1/2, equivalently α0→ 0, all massive modes of a string theory decouple and a good description is given by an effective field theory comprising only the massless excitation. Since the string length goes to zero in this limit strings in massless states are essentially pointlike and the

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16 CHAPTER 2. IIB SUPERSTRINGS AND BRANES quantum theory will correspond to a local quantum field theory.

The effective field theories of the five superstring theories are the five supergravity (SUGRA) theories in 10 dimensions. The name of each SUGRA coincides with that of the superstring theory it is the effective theory of (e.g., IIB SUGRA is the effective theory of IIB superstrings). Supergravities are supersymmetric theories containing general relativity, and are obtained by extending local Poincar´e invariance to in-clude local supersymmetry. Just like Einstein gravity, they are nonrenormalizable, reflecting their origin as effective theories. As field theories, they are considerably simpler than general strings to find background solutions to; therefore we will make extensive use of the supergravity approximation in the context of holography. 10D SUGRAs are perhaps easier to introduce starting instead from the unique 11D SUGRA. The field content of 11D SUGRA is as follows (we also note the number of physical polarizations):

Bosons Graviton g(M,N ) 44 3-form C3 = _3!1CM N LdxM ∧ dxN ∧ dxL 84

Fermions Gravitino Majorana ψM 128

As required by supersymmetry, the number of on-shell boson and fermion states are equal. These states form an irreducible supermultiplet, a gravity multiplet.

Upon dimensional reduction on a circle, in 10D these fields decompose into those of type IIA SUGRA:

NSNS Graviton g(µ,ν) 35 Kalb-Ramond B2 = 1₂Bµνdxµ∧ dxν 28 Dilaton φ 1 RR 1-form C1 = Cµdx µ ₈ 3-form C3 = _3!1Cµνρdxµ∧ dxν∧ dxρ 56

NSR, RNS Two gravitinos Weyl-Majorana ψ

(L)

µ , ψµ(R) 56 + 56

Two dilatinos Weyl-Majorana λ(L), λ(R) 8 + 8

where we have matched the fields with the massless modes of the four IIA string sectors. In fact, these fields are just the ground states defined in (2.29), decomposed in irreducible representations of SO(8). For example, the NS-NS ground Gµν =

˜_bµ†

1/2|0iN S⊗ b ν†

1/2|0iN S (reintroducing unphysical polarizations) is a spacetime

rank-2 tensor, decomposable as a symmetric form, an antisymmetric form, and a trace, as in

8V ⊗ 8V = 35 + 28 + 1 (2.38)

These are respectively the graviton, the Kalb-Ramond field, and the dilaton. The fields from the other sectors result accordingly from the decompostions:

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2.2. TYPE II SUPERGRAVITY AND D-BRANE CONTENT 17

N SR 8V ⊗ 8R= 56 + 8L (2.39)

RN S 8L⊗ 8V = 56 + 8R (2.40)

RR 8L⊗ 8R= 56 + 8V (2.41)

where the three inequivalent SO(8) irreps 8V, 8L, 8R are respectively the vector

and negative and positive chirality Weyl-Majorana representations.

We note the two gravitinos and dilatinos are of opposite chirality, as do the RNS and NSR sectors of IIA superstrings.

Obviously, again we find that the total bosonic states are 35+28+1+8+56 = 128 and the fermions 2 · (56 + 8) = 128. There is therefore a match both with the number of degrees of freedom of 11D SUGRA and with supersymmetry. IIA SUGRA in particular is a theory with N = (1, 1) SUSY, meaning there are two Weyl-Majorana SUSY generators of opposite chirality.

We will mainly be interested, however, in type IIB SUGRA, which is not obtainable from dimensional reduction, and is the effective field theory for IIB strings. The field content is as follows:

NSNS Graviton g(µ,ν) 35 Kalb-Ramond B2= 1₂Bµνdxµ∧ dxν 28 Dilaton φ 1 RR 0-form C0 1 2-form C2 = _2!1Cµνdxµ∧ dxν 28 4-form C4= _4!1Cµνρσdxµ∧dxν∧dxρ∧dxσ 35

NSR, RNS Two gravitinos Weyl-Majorana ψ

(1)

µ , ψ(2)µ 56 + 56

Two dilatinos Weyl-Majorana λ(1), λ(2) 8 + 8 The 4-form has its physical polarizations halved from 8₄ = 70 to 35 by the intro-duction of a constraint we will specify shortly.

IIB SUGRA has N = (2, 0) supersymmetry, with two equal-chirality Weyl-Majorana generators. It is therefore a chiral theory. While IIA is non-chiral and thus auto-matically anomaly free, IIB as a QFT has the potential to develop a gravitational anomaly due to the chiral fermionic sector. It is however found that the total anomaly miraculously cancels with the listed field content [15]. This cancellation is obvious in light of the fact that IIB SUGRA is the effective field theory to IIB superstrings, which are not just free from anomalies but from all UV divergences.

2.2.1 Gauge invariance of RR fields

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B2 → B2+ dΛ1 Cp → Cp+ dΛp−1− H3∧ Λp−3 (2.42)

for any set of arbitrary p-forms Λp, leaving invariant the field strengths:

H3:= dB2

Fp+1:= dCp− H3∧ Cp−2

(2.43)

Where Cp, Λp with p < 0 is set to 0. Now, intuitively, the RR form Cp couples to

(p − 1)-dimensional objects ((p − 1)-branes) through an interaction term of the type

Sint=

Z

Wp−1

Cp (2.44)

integrated over the p-dimensional worldvolume Wp−1, a sensible generalization of the

coupling of the EM potential to a charged particle. In particular, as it was already established that it is possible for open strings to end on D-branes, we would like to investigate for which values of p a certain string theory admits stable D(p − 1)-branes - certainly, if they have a conserved charge under a gauge field then they are protected from decay. Therefore the set of RR fields in a superstring theory determines the list of stable D-brane dimensionalities.

The above is an electric coupling of the (p − 1)-brane to Fp+1. The coupling

how-ever could also be magnetic, electric-magnetic duality being implemented in general through Hodge duality. We define Fp for additional values of p > 5 (odd for IIA,

even for IIB) through

Fp =e?F10−p (2.45) (Note that for the IIB F5 this would be actually a constraint, precisely the one that

acts on C4 to reduce its degrees of freedom; F5= ∗F5 however is not the exact form

of the constraint, as will be clarified in section 2.3). The new field strengths can then be locally trivialized as of (2.43) and so we end up with a complete set of potentials C0, . . . C8 for IIB and C1. . . C9 for IIA. The duality between potentials would act as

Cp ↔ C8−p, and if D(p − 1)-branes couple electrically to Cp, then D(7 − p)-branes

couple magnetically to it, that is to say electrically to C8−p.

Therefore, the magnetic dual to a Dp-brane is a D(6 − p)-brane. So the definitive list of stable D-branes in type II string theories along with the RR fields they are charge under is given by:

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2.3. ACTION FUNCTIONAL FOR IIB SUGRA 19 D(−1) C0 F1 D0 C1 F2 D1 C2 F3 D2 C3 F4 D3 C4 F5 D4 C5 F6 D5 C6 F7 D6 C7 F8 D7 C8 F9 D8 C9 F10 D9 C10 0

where shaded entries are for IIB, and unshaded for IIA, and arrows represent electric-magnetic duality. We comment on a few apparent anomalies.

• The IIB D(−1)-brane would have a 0-dimensional worldvolume, hence a single event. These type of branes are therefore actually instantons. They couple to the C0 potential, which has axionic character.

• IIA D8-branes and the C9 potentials do not have duals, yet D8-branes can be

shown to exist and to couple to the F10 field strength, as was first noted in

[31]. However the actionR F10∧ ?F10implies d ? F10= 0 which means F10is a

constant. Therefore there are no additional physical degrees of freedom from C9.

• Space-filling D9 branes can be introduced in IIB superstrings, but the equation of motion of the C10 form implies only special arrangements of D9s and

anti-D9s are allowed - this was first appreciated in [32].

2.3 Action functional for IIB SUGRA

There is a considerable obstacle to a covariant (i.e. explictly supersymmetric) for-mulation of type IIB supergravity in the self-duality constraint for the field strength 5-form F5. We will take the common path of formulating the Lagrangian theory

ignoring the constraint (and thus in excess of bosonic polarizations with respect to an explicity supersymmetric theory) and then imposing self-duality by hand after deriving the equations of motion. Therefore the action will not be supersymmetric itself, while the Euler-Lagrange equations augmented with the constraint will be. This issue becomes very problematic on quantization, but, as we will prove, classical supergravity will be sufficient for our purposes.

Actually, for the purpose of building classical solutions, where spinor fields vanish anyway, the fermionic sector of the action will not be important. After introducing

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20 CHAPTER 2. IIB SUPERSTRINGS AND BRANES the “string length” `s through6

`s:= 2π

√

α0 _(2.46)

the bosonic sector is as such:

SIIB,B = SN S+ SR+ SCS (2.47)

where SN S is the action relevant to the fields originally from the superstring NS-NS

sector: SN S = 2π `8 s Z d10x√−g e−2φ R + 4∂µφ∂µφ − 1 2|H3| 2 (2.48)

where the p-form norm is |ω|2 = ω ∧ ?ω. Then SR is for R-R fields, essentially just

kinetic terms for the A forms:

SR= − 2π `8 s Z d10x√−g |F₁|2+ | ˜F3|2+ 1 2| ˜F5| 2 (2.49) ˜ F3 := F3− C0H3 (2.50) ˜ F5 := F5− 1 2C2∧ H3+ 1 2B2∧ F3 (2.51) And finally we supplement with a Chern-Simons type term:

SCS = − 2π `8 s Z C4∧ H3∧ F3 (2.52)

note the untilded F3. This is evidently a purely topological term.

The self-duality constraint is then imposed in terms of the modified field strength ˜

F5

˜

F5 = ∗ ˜F5 (2.53)

The action presented above is in what is known as the “string frame”. It becomes convenient in many occasion to switch to an alternative formulation through a field redefinition, to move to the “Einstein frame”. The change is

g_µνEF = e−φ/2g_µνSF (2.54)

6_{Care should be taken with inequivalent convention with the definition of the string length in}

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2.4. D-BRANES 21 and the action in terms of the new metric is [4]

`8 s 2π SEF = Z d10x√−g R −1 2∂µφ∂ µ_{φ −}1 2e −φ_|H 3|2 −1 2 Z d10x√−g e2φ|F₁|2_{+ e}φ_{| ˜}_F 3|2+ 1 2| ˜F5| 2 −1 2 Z C4∧ H3∧ F3 (2.55)

The advantage of this picture is the canonical Einstein-Hilbert and dilaton terms; by converse exponential couplings of the dilaton with the form fields are introduced.

2.4 D-branes

2.4.1 D-brane action

D-branes appear as nonperturbative objects in string theories. They are themselves dynamical and the dynamics are modeled in the string perturbative regime by an action functional [15]. To formulate the action, we introduce coordinates σa on the (p+1)-dimensional worldvolume Wpand functions Xµ(σa) describing the embedding

of Wp in spacetime as ι : σα 7→ Xµ(σa). It is then tempting to choose as bosonic

action the obvious generalization of the Nambu-Goto action:

SDp= −µDp

Z

Wp

dp+1σp− det ι∗_{(G) = −µ}

Dpvol Wp (2.56)

the notation ι∗(T ) denotes the pull-back of a spacetime tensor to the worldsheet. For example, ι∗(G) is the induced metric hab = ∂aXµ∂bXνGµν. µDp would be the

D-brane tension. The insight of (2.56) is correct, but incomplete; firstly it does not include a coupling with possible background fields, but most importantly it does not account for all of the open string modes living on the worldvolume.

In fact, if the quantization procedure we performed for the open string is repeated with the endpoints constrained to a Dp-brane (i.e., with 9 − p Dirichlet and p + 1 Neumann conditions) then one finds among the massless bosonic states both scalar (from the worldvolume point of view) fields Xp+1,...,9 which indeed correspond to motion of the D-brane in the transverse directions, but also, as we have seen, a massless vector potential mediating a U (1) gauge theory confined to the D-brane volume. Therefore, it is directly from the modes of open strings that it is possible to deduce the brane is itself a dynamical object, as its position is related to VEVs of open-string scalars. Enforcing this idea that the D-brane dynamics should be encoded in those of the open-string modes, D-branes should always host at least a

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22 CHAPTER 2. IIB SUPERSTRINGS AND BRANES U (1) gauge theory on them.

The bosonic part of the Dp-brane action is then found to be

SDp = − µDp Z W dp+1σe−φp− det (ι∗_{(g − B} 2) − 2πα0F ) (2.57) + µDp Z W " ι∗ X k Ck ! ∧ e2πα0F −B2 _{∧ (1 + O(R}2₎₎ # p+1 (2.58)

Where µDp can be fixed as

µDp= α0 −

p+1

2 (2π)−p (2.59)

and F is the field-strength 2-form of the U (1) gauge theory.

The first line (2.57) is the Dirac-Born-Infeld action and generalizes the Nambu-Goto action; Setting only B = 0, φ = const and expanding SDBI in powers of α0:

SDBI = − µDp gs Z W dp+1σ√−h + α 0−(p−3)/2 4gs(2π)p−2 Z W dp+1σ√−h F_µνFµν + . . . (2.60)

the first term is the direct generalization of the Nambu-Goto action, allowing us to identify the Dp-brane tension TDp=

µDp

gs . The second is a Yang-Mills action for the

U (1) gauge field, restricted to the worldvolume.

It becomes clear D-branes carry a mass per unit p-volume TDp ∼ g

−1

s and are thus

nonperturbative in terms of the string coupling. Note also the Maxwell action is weighted by g_s−1, in agreement with what previously found for the relation between open and closed string couplings: gY M ∼ g0 ∼ gs1/2. To be more exact, comparing

with the canonical Maxwell action one can deduce

g_{Y M}2 = 2πgs (2.61)

The second line (2.58) is a Chern-Simons type term coupling the brane to the RR potentials. The sum over k only spans odd or even respectively for IIA or IIB, and the [ ]_p+1 notation means the p + 1-form component must be selected so as to define a meaningful integral. We note that in vanishing curvature, and expanding in 2πα0F − B2, the physical interpretation becomes less obscure:

SCS = µP Z W Cp+1+ µP Z W Cp−1∧ (2πα0F − B2) + O(F2) (2.62)

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2.4. D-BRANES 23 the zeroth order in F . A Dp-brane is therefore also understood as a localized charge for the Fp+2field. Higher order terms mean a coupling with the lower RR potentials

and are due to nontrivial F configurations which induce lower-dimensional D-brane charges localized inside the Dp-brane.

We touch briefly upon the easy generalization of the above action to the case of N coincident Dp-branes, a “stack”. We imagine first taking 2 separated parallel D-branes (1 and 2) and bringing them closer together. Normally, open string modes stretching from 1 to 2 are suppressed by the increase in mass squared due to the minimum elastic energy to span the inter-brane distance. In fact, the open string spectrum is found to be identical except for the constant shift in the mass-shell condition [4]: ∆M2= T2 9 X i=p+1 (X₁i − X₂i)2 (2.63)

If then the branes are brought to coincide, this shift vanishes and one must admit an additional 1 → 2 sector of massless states has been created. In general, with N coincident Dp-branes there will be an N × N matrix of massless sectors indexed by a, b = 1, . . . , N marking the starting and ending brane (Chan-Paton indices). This means then a matrix of gauge vectors Aab generating U (N ) gauge transformations;

this U (N ) group is nothing else than transformations mixing the N identical, coin-cident D-branes with eachother. Therefore they act on Chan-Paton indices in the defining representation.

Then, it is clear the fields Φi_ab := (X_ai − Xi

b), parametrizing relative D-brane

trans-verse position, transform in the adjoint of the gauge group. The action of separating again the D-branes is then interpreted in this point of view as a Higgsing of the gauge group by these scalars; when the stack of N branes splits into two groups of N1 and

N2 branes which are separated, this corresponds to the relevant Φ fields acquiring a

VEV, breaking part of the gauge group to the corresponding group for two separate stacks

U (N ) → U (N1) × U (N2) (2.64)

and the gauge fields corresponding to the broken generators, which gain a mass through the Higgs mechanism, are in fact modes of strings stretching between the two stacks, so that the Higgsed mass can also be viewed as the elastic energy from (2.63).

The salient point in any case is the extension of the gauge group from U (1) to U (N ). Essentially, the F2 term (and higher) in (2.60) must be supplemented with gauge traces.

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2.4.2 D-branes as supergravity solitons

As D-brane carry mass, a dual description of D-branes in terms of the warped space-time they produce should be possible. These spacespace-times indeed appear as solitonic solutions in supergravity generalizing the usual four-dimensional black hole metrics, under the name of p-branes or black branes. In particular, as Dp-branes are (for certain values of p) stable objects, stabilized by Fp+2charge, their supergravity

des-cription must be a zero-temperature state, unable to lose mass to Hawking radiation, and thus extremal.

These solutions are constructed in complete analogy to the derivations of usual general relativity black holes, by inserting an ansatz with the desired symmetries into the supergravity action, and in fact are direct generalizations of the extremal Reissner-Nordstr¨om hole. We provide a succint review of the derivation and result, following for example [18]. Since the p-brane is stabilized by its charge under the Cp+1potential or the Fp+2field strength, it can be assumed all the other form fields

vanish, including H3. Moreover the fermion fields vanish for classical solutions.

Therefore the Einstein frame IIB action reduces to

SIIB = 2π `8 s Z _√ −g R − 1 2(dφ) 2₋ 1 2ηe 3−p 2 φ|F_p+2|2 (2.65)

if p = 3, the self-duality constraint F5 = ˜F5 is to be imposed after finding the

Euler-Lagrange equations, and η = 2; otherwise η = 1.

It is clear one can assume the black p-brane has Rp+1o SO(1, p) × SO(9 − p) symmetry, so that we can introduce a set of longitudinal coordinates x0,...,p and transverse coordinates yp+1,...,10 such that the dependence of all fields components in this coordinates is reduced to the single radial transverse variable r2= ~y · ~y. The most general Einstein frame metric with these symmetries is then

ds2= H_p−1/2dx · dx + H_p1/2dy · dy (2.66) where the warp factor Hp(r) is a function of r only, and dx · dx and dy · dy =

dr2_{+ r}2_dΩ

8−p are respectively the Minkowski and Euclidean metrics. Analogously,

the dilaton φ is also a function of r and the form potential must take the form

C012...p = C(r) (2.67)

After variation of the action (2.65) and insertion of the described ansatz in the resulting equations of motion (plus simplifications in the p = 3 case since |F5|2 =

F5∧ ∗F5 = F5∧ F5= 0), one is left with a differential equation for Hp. It is found,

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2.4. D-BRANES 25

∇2_H

p = 0 (2.68)

where ∇2 is the flat space Laplacian, hence a linear equation. This traces back to the fact that multiple extremal black holes are non-interacting, as the gravitational and electrostatic forces cancel - the same holds for D-branes in supergravity. This makes it possible to construct exact solutions with multiple branes by superposition. In any case, the linear equation has (taking into account the boundary condition of asymptotic flatness Hp(∞) = 1) a simple solution as (for p < 7):

Hp(r) = 1 +

Rp

r 7−p

(2.69) for some Rp to be determined. Exploiting the equations of motion one finds also

eφ(r) = gsHp(r)(3−p)/4 (2.70)

where gs= eφ(∞) is the background value of the string coupling, and

C01...p= Hp−1− 1 (2.71)

⇒ F_p+1= d(H_p−1) ∧ dx0∧ . . . ∧ dxp (2.72)

which completes the specification of the class of solutions. Now Rp is fixed by

matching the gravitational flux to the mass (per unit longitudinal volume) of the D-brane. In fact, we are free to consider a set of N coincident Dp-branes, a “stack”, an easy generalization which will prove very important in the context of AdS/CFT. The exact dependence is

(Rp)7−p= (4π)5−p2 Γ 7 − p 2 α07−p2 g_sN (2.73)

While this result was derived from the IIB action and so for p = 1, 3, 5, it actually applies identically for IIA p = 0, 2, 4, 6 branes7. We single out from this the self-dual case of the D3 brane which displays a uniform value for the dilaton. In general instead from (2.70) and the warp factor (2.69) we have the near-horizon behaviour

e4φ ∼r R

(3−p)(p−7)

(2.74) which means the local string coupling diverges for p < 3 and vanishes for p > 3.

7

It should be remarked that similar D7 and D8 solutions are possible, but that the behaviour of the warp factor is respectively logarithmic and linear in r, and so asymptotic flatness cannot be enforced.

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26 CHAPTER 2. IIB SUPERSTRINGS AND BRANES p = 3 acts as a middle ground between these cases.

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Chapter 3

D3-brane stacks on Calabi-Yau

cones

One of the most essential ingredients for the conception of the idea of holography was the fact that a stack of coincident D3-branes naturally features a 4D gauge theory on their world-volume, where the 4D fields emerge from the modes of open strings stretching between them. In the simplest and most famous example, a stack of N D3-branes is placed in otherwise Minkowski R1,9; the corresponding field theory is the maximally supersymmetric Yang-Mills in four dimensions (N = 4 SYM ). Setting the stack on a different background geometry instead gives rise to a large family of different field theories; a particularly interesting subset is given by space-times of the form

M = R1,3× X6, (3.1)

where the R1,3 is parallel to the branes (and must be identified with the field theory spacetime) and X6 is a 6-dimensional Calabi-Yau cone over a compact 5-fold base

Y5. By X6 being a cone it is meant there exists a conical radial coordinate r such

that the metric on X6 is of the form

ds2 = dr2+ r2ds2₅ (3.2) With ds2₅ the metric on Y5.

In this language, the N = 4 SYM example above corresponds to X6 = R6 = C3,

which is (trivially) a cone over Y5 = S5. This is the only case where X6 turns out to

be smooth; in general it will feature a conical singularity in the origin. The moti-vation for jumping so hastily to a generalization as radical as a singular spacetime, instead of other smooth spacetimes, is that the latter will not actually introduce any novel features. As will be clarified in a holographic context (see for example

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28 CHAPTER 3. D3-BRANE STACKS ON CALABI-YAU CONES section 4.4), the flow towards the IR of the field theory will actually correspond to “zooming in” on the D-branes, and any smooth spacetime will converge to flat R6 in this limit. Only a genuine singularity is going to introduce any novel behaviour in the IR field theory, and conical defects are a well-known example of singularities on which string theory is known to be formulable [11].

Non-trivial choices for the base will typically yield theories with reduced (even mi-nimal) supersymmetry, which are considerably more challenging to study.

In this chapter, we will first describe some general features of the theories resulting from the placement of D3-branes on these conical backgrounds. Then, we will con-centrate in particular on a specific chain of field theories starting from the simplest example of N = 4 SYM and ending up on the Y2,0 _{theory, the study of which is the}

main objective of this work.

3.1 Superconformal field theory

We now provide a short introduction to 4D conformal field theories, their supersym-metric variants, and the relevant terminology.

Consider flat spacetime of dimension d and signature p, q, with p + q = d. Take a coordinate chart xµ in which the metric takes the standard form ηµν. Conformal

transformations are defined to be diffeomorphisms xµ→ x0µ which leave the metric unchanged in form up to an x dependent scalar function (a conformal factor):

g_µν0 (x0) = Ω(x)ηµν. (3.3)

These maps form evidently a group, which is known as the conformal group CO(p, q). We will mainly be interested in CO(1, d − 1), but most of what we will now show applies in general signatures.

In d > 2, the conformal group will turn out to be a finite-dimensional Lie group, of which we specify now the connected component. An obvious subgroup is maps that leave the metric unchanged, so Poincar´e transformations, with generators Pµ and

Jµν. A second easy guess is the subgroup with constant conformal factors, that is

scale transformations or dilations

xµ→ λxµ ηµν → λ−2ηµν (3.4)

whose generator is called D. To generate the whole conformal group a final class of transformations must be introduced, special conformal transformations, generated by Kµ and with finite action1

1

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de-3.1. SUPERCONFORMAL FIELD THEORY 29 xµ→ x µ_{− b}µ_x2 1 − 2bν_x ν + b2x2 (3.5) Together, Pµ, Jµν, D and Kµ generate the connected component of the conformal

group in d dimensions. The extension of the Poincar´e algebra to the conformal one is characterized by the following additional commutators (using hermitian generators)

[Jµν, Kρ] = 2iηρ[µKν] (3.6)

[Jµν, D] = 0 (3.7)

[D, Pµ] = iPµ (3.8)

[D, Kµ] = −iKµ (3.9)

Equations (3.6) and (3.7) just confirm Kρ is a vector and D is a scalar. (3.8) and

(3.9) instead state that Pµand Kµare respectively raising and lowering operators for

D. It is worth of notice that this group CO(1, d−1) is actually SO(2, d), the Lorentz group in mixed signature (2, d). This can be shown by combining the generators in

JM N =    Jµν (Kµ− Pµ)/2 −(Kµ+ Pµ)/2 (Pµ− Kµ)/2 0 D (Kµ+ Pµ)/2 −D 0    (3.10)

and then it can be verified that JM N satisfy the algebra of so(2, D). This

equiva-lence will be relevant when we will introduce AdS/CFT, since SO(2, D) is also the isometry group of AdSD+1.

A quantum field theory which has the conformal group as symmetries is called a conformal field theory (CFT). In such a theory, particles lie in irreducible represen-tations of the conformal group; since the mass P2 is not a Casimir for the whole group, it becomes useful to replace it with more relevant quantum numbers. Con-sider the dilation operator: in the quantum theory it will be represented by

D = −i(xµ∂µ+ ∆) (3.11)

where ∆ gives the intrinsic scaling dimension of a field, which will in general trans-form as φ(x) → λ∆φ(λx). ∆ is therefore a good quantum number. Considering the role of P and K as ladder operators, changing the conformal dimension by ±1, we can deduce states will come in multiplets of ever-increasing dimension ∆₍₀₎+ n, n ≥ 0, and that the lowest-dimension state will be annihilated by Kµ. Fields in the

kernel of Kµ will be called primary, and others, obtained by applying powers of Pµ

nominator can vanish. Indeed, these more naturally act on the conformal compactification Rp,q_,

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30 CHAPTER 3. D3-BRANE STACKS ON CALABI-YAU CONES (hence, derivatives) will be called descendants.

A primary field is then identified by its conformal dimension and its representation under the Lorentz group, so, now specializing to D = 4, by quantum numbers (∆, jL, jR). We recall Lorentz irreps are indexed by two half-integers (jL, jR), for

example: (0, 0) is a scalar, (1₂, 0) and (0,1₂) are left/right Weyl spinors, (1₂,1₂) is a vector, and so on.

A classically conformal field theory very often fails to be conformal when quantized. This happens because the dilation symmetry is anomalous. Classical scale invariance clearly implies all couplings are adimensional; in the quantum theory these couplings gi will run under renormalization with a corresponding β function, as in

dgi d ln µ =: β

i_{(g) .} _(3.12)

The dependency of the running coupling on the energy scale, or equivalently the cre-ation of a mass scale by dimensional transmutcre-ation, means the conformal symmetry is spoiled2. This happens for example in quantum chromodynamics, a classically con-formal theory with a scale anomaly giving rise to the ΛQCDmass scale, or quantum

electrodynamics where the scale is at the Landau pole. Since the Noether current corresponding to dilations is the trace of the energy-momentum tensor, the anomaly will be detectable by the appearence of a nonzero matrix element hTµµi ∝ β(g) 6= 0.

Only if all the β functions vanish identically, i.e. if the theory is finite, is quantum conformal invariance guaranteed. We will encounter an example of such a theory in section 3.3. Otherwise the theory will only be conformal for specific values of the gi at which all the β functions vanish, that is to say at fixed points. In general a quantum field theory will flow under renormalization from a non-conformal point towards an attracting IR fixed submanifold, the locus of βi(g) = 0 , called the conformal manifold.

An important point is that after the theory has regained its classical conformal symmetry after converging through RG flow to an IR fixed point, the quantum scaling dimensions ∆ of operators will not coincide with the original value they had in the classical theory, the canonical dimension ∆0. They will be modified by

quantum corrections that add an anomalous dimension

∆ = ∆0+ γ(g∗) , γ(g) = − 1 2 d ln Z d ln µ, (3.13) 2

We are here using conformal and scale (i.e. dilation) invariance interchangeably, but they are not identical. Conformal symmetry obviously includes dilations, but scale invariance + Poincar´e does not generate the whole conformal group, as special conformal transformations are independent. Scale invariant but not conformal theories are known explicitly [2], but they are rare. We will work with the assumption dilation-invariant ⇒ conformal.

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3.1. SUPERCONFORMAL FIELD THEORY 31 where√Z renormalizes the wavefunction3, and g∗ are the values of the couplings at

the conformal fixed point.

Having introduced the extension of the Poincar´e group to SO(2, 4), we would like to press this further to include supersymmetry. Supersymmetry is implemented by adding N ≤ 4 Weyl supercharges QA, Q_A (A = 1, . . . , N ) to generate the super-Poincar´e supergroup ISO(1, 3| N ). The superconformal group SO(2, 4| N ) is then the minimal supergroup containing both. The first important feature is that a second set of supercharges SA, S

A

must be introduced to close the algebra, since

[Kµ, QA] = −σµS A

, [Pµ, SA] = QAσµ; (3.14)

so that superconformal symmetry SO(2, 4| N ) involves twice as many supercharges as normal supersymmetry for a given N . Another relevant excerpt from the table of commutators (which we do not reproduce in full) states QA and SAare also ladder

operators for dilations,

[D, QA] = i 2Q A_, _{[D, S} A] = − i 2SA, (3.15) raising and lowering the dimension ∆ by ±1/2. In a superconformal field theory (SCFT) we then expect multiplets of dimension ∆ = ∆0+ n₂. Primary operators

must now be annihilated by both Kµand SA, and are classified again by dimension

and spin (∆, jL, jR) but also by the U (1) × SU (N ) R-symmetry quantum numbers

(R, r) (r denoting a generic irrep of SU (N )). Then, by acting with the raising operator QA charges one can reconstruct a finite-dimensional supermultiplet, as in normal supersymmetry. Instead, powers of Pµ reconstruct the infinite ladder of

derivatives forming an infinite-dimensional representation of the conformal group; these can be recombined into a field by Taylor expansion. In conclusion, an infinite-dimensional representation of the superconformal group can be organized into a superfield

Φ...(xµ, θA, θ A

) (3.16)

where . . . stands for Lorentz and R-charge-SU (N ) indices for the primary.

Actually, not all values for ∆ are allowed in a quantum theory. Imposing physi-cal states have non-negative norm (i.e., unitarity) results in lower bounds for the quantum scaling dimension [26],

3_{It should be noted some authors prefer to define γ = −}d ln Z

d ln µ. In addition, Z is generally a

matrix that mixes different fields together under RG flow; however in all of the cases considered in this work this can be ignored, since only fields in the same gauge representation and with the same spin could ever mix, and those will always turn out to be connected by a flavour symmetry.

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32 CHAPTER 3. D3-BRANE STACKS ON CALABI-YAU CONES

∆ ≥ f (j1, j2) ; (3.17)

moreover, since violation of the bound results in negative norms, by continuity when it is saturated zero-norm states appear. This correspond in general to a shortening of the multiplet, which becomes constrained to be annihilated by a polynomial of Pµ. For example, for scalar fields

∆ ≥ 1 (3.18)

and ∆ = 1 iff ∂µ∂µΦ = 0, that is, Φ is free. In the spin-1 case, ∆ ≥ 3 and equality

holds only if the field is a conserved current (∂µJµ= 0); for spin-2 ∆ ≥ 4 and ∆ = 4

only if ∂µTµν = 0, that is to say Tµν must be the stress-energy tensor. This result

implies in particular that conserved operators must have fixed, canonical dimension and so are not renormalized.

In N ≥ 1 SCFTs, more interesting unitarity bounds can be introduced by extending the above reasoning to include superconformal symmetries. Introducing the U (1) R-charge symmetry (and normalizing such that RQ = 1), one is led to bounds of

the type

∆ ≥ f (j1, j2, R) (3.19)

depending also on the R-charge of the superfield. Saturation corresponds to the appearance of ghosts and a shortening of the multiplet, which is annihilated by a polynomial of Pµand Q. In particular, for scalars

∆ ≥ 3

2R (3.20)

and equality holds iff Dα˙Φ = 0, i.e. if the superfield is chiral. Thus, chiral fields will

satisfy

∆ = 3

2R . (3.21)

3.2 Features of D3-brane on Calabi-Yau cones

We consider a stack of N D3-branes on a ten-dimensional background of the form R1,3× X6. The branes are parallel to the R1,3 (which can be identified with the

worldvolume) and are essentially points from the point of view of the 6D manifold X6. Since, as it was anticipated, there is an interest in having the D-branes probe a

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3.2. FEATURES OF D3-BRANE ON CALABI-YAU CONES 33

ds2₆ = dr2+ r2ds2₅ (3.22) If Y5 = S5 with the unit round metric then the cone is X6 = R6 and one returns to

the flat case. Therefore we include this as a trivial example of a cone.

In addition, we must require that the cone be Ricci-flat, so that it satisfies the supergravity equations of motion in vacuum. This is equivalent to Y5 being Einstein

of positive curvature, as we now show. ds2₆ is conformally equivalent to the canonical metric on a cylinder over Y5, as evidenced by the reparametrization φ = ln r:

ds2₆= e2φ dφ2+ ds2₅ ; (3.23) recalling the transformation law of the Ricci tensor in n dimensions under conformal rescalings: R0_ij = Rij − (n − 2) (∇i∂jφ − ∂iφ∂jφ) + ∇2φ − (n − 2)∇kφ∇kφ gij, (3.24)

and noting that for the cylinder (which has a product metric) the restriction of Rij

to Y5 indices gives Y5’s own Ricci tensor R(5)_ij , we obtain

R(5)_ij = 4g(5)_ij . (3.25) A manifold with Rij = Λgij, with Λ a constant, is called Einstein.

Also, we require that X6 be K¨ahler, with the Ricci-flat metric gij being the K¨ahler

metric. This restrictive property is necessary [16] for the field theory to have at least N = 1 supersymmetry.

Indeed, bein K¨ahler and Ricci-flat implies that X6 is a Calabi-Yau manifold, thus of

restricted holonomy ⊂ SU (3). More restricted holonomy results in enhanced super-symmetry; in particular if the holonomy group is contained in SU (2) then the gauge theory will have N = 2, and if the holonomy is trivial (i.e. X6= R6) then the

super-symmetry will be maximal, N = 4. Intuitively, this is because supersymmetries of X6 carry over as rigid supersymmetries of the field theory; this fact will be explained

more rigorously in the context of holography, however. An Einstein manifold Y5 such

that the corresponding cone X6 is Calabi-Yau is called Sasaki-Einstein.

Independently of the background, theories resulting from D3-brane stacks will always be gauge theories, as gluons mode will always be present. In particular, the gauge group will be a product of U (N ) factors (nodes); the number of gauge factors is related to the topology of the cone as it is its Euler characteristic χ [24].

Holographic eﬀective ﬁeld theories: a case study

Universit`

a degli Studi di Padova

Dipartimento di Fisica e Astronomia

Corso di Laurea Magistrale in Fisica

Holographic effective field

theories: a case study

Contents

Chapter 1

Introduction

Chapter 2

IIB superstrings and branes

2.1

Superstring theory

2.2

Type II supergravity and D-brane content

2.3

Action functional for IIB SUGRA

2.4

D-branes

Chapter 3

D3-brane stacks on Calabi-Yau

cones

3.1

Superconformal field theory

3.2

Features of D3-brane on Calabi-Yau cones